I just wanted to be able to have all of the text selected when the user clicks inside the input box.  The WebTextEditor has a property that makes this easy, but couldn't find anything built into the WebDropDown that could accomplish this... even with the CSOM.

This bit of code does the trick:

    <script type="text/javascript" id="igClientScript1">
<!--

        function wddRouteBookID_Focus(sender, eventArgs) {
            ///<summary>
            ///
            ///</summary>
            ///<param name="sender" type="Infragistics.Web.UI.WebDropDown"></param>
            ///<param name="eventArgs" type="Infragistics.Web.UI.DropDownControlEventArgs"></param>
            sender._elements.Input.select();
        }// -->
    </script>

Not crazy about using one of the internal functions, but it is the only way I could figure how to do this.